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The hyperfine structures of the $2,^3!S_1$ states of the $^6$Li$^+$ and $^7$Li$^+$ ions are investigated theoretically to extract the Zemach radii of the $^6$Li and $^7$Li nuclei by comparing with precision measurements. The obtained Zemach radii are larger than the previous values of Puchalski and Pachucki [href{https://link.aps.org/doi/10.1103/PhysRevLett.111.243001}{Phys. Rev. Lett. {bf 111}, 243001 (2013)}] and disagree with them by about 1.5 and 2.2 standard deviations for $^6$Li and $^7$Li, respectively. Furthermore, our Zemach radius of $^6$Li differs significantly from the nuclear physics value, derived from the nuclear charge and magnetic radii [href{https://link.aps.org/doi/10.1103/PhysRevA.78.012513}{Phys. Rev. A {bf 78}, 012513 (2008)}], by more than 6 sigma, indicating an anomalous nuclear structure for $^6$Li. The conclusion that the Zemach radius of $^7$Li is about 40% larger than that of $^6$Li is confirmed. The obtained Zemach radii are used to calculate the hyperfine splittings of the $2,^3!P_J$ states of $^{6,7}$Li$^+$, where an order of magnitude improvement over the previous theory has been achieved for $^7$Li$^+$.
Determination of nuclear moments for many nuclei relies on the computation of hyperfine constants, with theoretical uncertainties directly affecting the resulting uncertainties of the nuclear moments. In this work we improve the precision of such met
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